Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,18,15)(12,16,20,22)(19,24,21,23)(26,32)(27,28)(30,40)(34,37);(1,2)(3,7)(4,5)(6,8)(10,13)(11,18)(16,22)(19,21)(25,26,31,32)(27,33,28,36)(29,30,39,40)(34,38,37,35);(2,5)(6,7)(12,20)(16,22)(19,21)(23,24)(25,27,31,28)(26,33,32,36)(29,37,39,34)(30,35,40,38);(2,5)(6,7)(9,11,14,18)(10,15,13,17)(12,21,20,19)(16,24,22,23)(25,28,31,27)(26,33,32,36)(29,37,39,34)(30,38,40,35);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,29)(26,30)(27,34)(28,37)(31,39)(32,40)(33,38)(35,36);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,17,18,15)(12,22,20,16)(19,24,21,23)(25,30)(26,29)(27,35)(28,38)(31,40)(32,39)(33,37)(34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10230135);